In race to shift services to web or mobile phones, the stakes are in the billions of dollars for traditional lenders and upstarts alike

By some measures, PayPal Holdings Inc. holds more customer money than all but 20 U.S. banks. What does that say about the future of banking?

At the end of March, customers of PayPal held more than $13 billion in accounts at the online-commerce company. Compared with the most similar type of bank deposit, the figure would put PayPal just behind TD Bank, the U.S. unit of Toronto-Dominion Bank, or Capital One Financial Corp., according to data provider S&P Global Market Intelligence.